Properties

Label 2-1890-63.47-c1-0-4
Degree $2$
Conductor $1890$
Sign $0.578 - 0.815i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 5-s + (−2.64 − 0.0270i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s + 0.288i·11-s + (−0.872 + 0.503i)13-s + (−2.30 + 1.29i)14-s + (−0.5 − 0.866i)16-s + (2.94 + 5.10i)17-s + (−1.59 − 0.919i)19-s + (−0.499 + 0.866i)20-s + (0.144 + 0.250i)22-s + 5.54i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.999 − 0.0102i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + 0.0870i·11-s + (−0.241 + 0.139i)13-s + (−0.615 + 0.347i)14-s + (−0.125 − 0.216i)16-s + (0.714 + 1.23i)17-s + (−0.365 − 0.210i)19-s + (−0.111 + 0.193i)20-s + (0.0307 + 0.0533i)22-s + 1.15i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487721005\)
\(L(\frac12)\) \(\approx\) \(1.487721005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (2.64 + 0.0270i)T \)
good11 \( 1 - 0.288iT - 11T^{2} \)
13 \( 1 + (0.872 - 0.503i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.94 - 5.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.59 + 0.919i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.54iT - 23T^{2} \)
29 \( 1 + (-4.00 - 2.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.24 - 1.29i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.62 - 2.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.10 + 5.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.67 - 4.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.50 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.98 + 5.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.37 - 9.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.46 - 2.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.75 + 9.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.84iT - 71T^{2} \)
73 \( 1 + (2.67 - 1.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.991 - 1.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.43 - 9.41i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.18 - 2.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.09 - 5.25i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.475131713873858638263845665678, −8.612916047339041697906799195046, −7.66935663013021996930704272135, −6.82917190484281767285164999931, −6.11161676969468012904936800168, −5.27991747083690621727415369602, −4.21353910176253991842500160938, −3.51828998949208536303713735280, −2.68375071730065201199831746196, −1.28889482100634173938220147197, 0.46350377529821749132091934949, 2.47875046158764016770223394502, 3.24889312018680913767994451259, 4.15608110973433168908521684985, 5.03789391432668055061525860505, 5.92582524686542957648109537915, 6.74575781180650458763261054519, 7.31599195150330286168503083829, 8.241556577339853851797325114808, 8.972579322437070715418998579198

Graph of the $Z$-function along the critical line